Transformations

Inverses

Extrema

Incr. & Decr.

Composition

What is an equation for the squaring function shifted up by 2?

f(x) = x^{2} + 2

What is an equation for the inverse of the function f(x) = 5x, if the inverse exists?

f(x) = x/5

What are the x-values of the relative extrema of the function f(x) = x^{3} – 3x^{2} + x + 1?

relative maximum at x = 0.18; relative minimum at x = 1.8

Using interval notation, state the intervals where the function f(x) = (2x – 3)^{2} is increasing and where it is decreasing.

decreasing on (–∞,1.5)U(1.5, ∞)

If f(x) = x^2 and g(x) = x – 2, what is the equation for f(g(x))?

f(g(x)) = (x – 2)^2

What is an equation for the absolute value function shifted to the left by 20?

f(x) = |x + 20|

What is the equation for the inverse of the function f(x) = 7x + 1, if the inverse exists?

f(x) = (x – 1)/7

What are the relative and absolute extrema of the function f(x) = x^{4} + x^{3} – 2x^{2} – x + 2?

Absolute minimum at x = –2.4; Relative minimum at x = 0.4; Relative maximum at x = –0.25

Using interval notation, state the intervals where the function f(x) = (2x – 3)^{2} – x^{2} is increasing and where it is decreasing.

decreasing on (1.19,1.90); increasing on (–∞,1.19)U(1.90, ∞)

If f(x) = x^2 and g(x) = x – 2, what is the equation for g(f(x))?

g(f(x)) = x^2 – 2

What is an equation for the cubic function stretched vertically by a factor of 4 and reflected across the y-axis?

f(x) = 4(-x)^{3}

What is the equation for the inverse of the function f(x) = x/2 + 15, if the inverse exists?

f(x) = 2(x – 15)

What are the relative extrema of the function f(x) = 0.05x^{3} – 0.8x^{3} + 0.4x + 1?

Relative maximum at x = 0.26; relative minimum at x = 10.4

Using interval notation, state the intervals where the function f(x) = 1/(x-5) is increasing and where it is decreasing.

Increasing on (-ထ,5)U(5,ထ)

If f(x) = 2x^2 – x and g(x) = x – 5, what is the equation for f(g(x))?

f(g(x)) = 2(x – 5)^2 – (x – 5)

What is an equation for the quadratic function shifted up by 5, to the right by 4, and reflected across the x-axis?

f(x) = – (x – 4)^{2} + 5

What is the equation for the inverse of the function f(x) = 2 – 3x^{2}, if the inverse exists?

The function has no inverse, it fails the horizontal line test.

What are the relative and absolute extrema of the function f(x) = 0.04x^{4} + 0.2x^{3} – 1.9x^{2} + 0.1x + 4?

absolute minimum at x = –7.1; relative maximum at x = 0.026; relative minimum at x = 3.3

Using interval notation, state the intervals where the function f(x) = 0.6x^{4} + 1.8x^{3} + x^{2} – x + 1 is increasing and where it is decreasing.

decreasing on (–∞,1.52)U(–1, 0.27); increasing on (–1.52, –1)U(0.27, ∞)

If f(x) = 2x^2 – x and g(x) = x – 5, what is the equation for g(f(x))?

g(f(x)) = 2x^2 – x – 5

A function, f(x)=-4(x+3)^{2}-5, is shifted up by 3, translated left 2 units, and vertically compressed by a factor of 1/2. What is the equation of the new function?

f(x) = -2(x+5)^{2}-2

What is the test to determine if a relation's inverse is a function?

How do you verify inverses algebraically?

Horizontal Line Test

F and G are inverses if f(g(x))=x and g(f(x))=x

What are the relative and/or absolute extrema of the function f(x) = 0.01(x – 5)^{4} – 0.1x^{3} – 0.4x + 4?

Specify if it is a relative or absolute maximum or minimum.

The function has an absolute minimum at x=18.88

Describe increasing and decreasing intervals. Your response must include the direction you read the graph, and an explanation of the change in x as it relates to the change in y.

Must include: Read the graph from left to right

Increasing: As x increases, y increases

Decreasing: As x increases, y decreases

If f(x) = 2x^2 – x, g(x) = x – 5, and h(x) = x/3, what is the equation for h(g(f(x)))?

h(g(f(x))) = (2x2 – x – 5)/3

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